PX284 - T4 - thin lens

for light travelling left-to-right:
- $u > 0 ⟹$ object to the left of the surface
- $v > 0 ⟹$ image to the right of the surface
now combining the two sides of a spherical lens, such that the image of the first (left) lens becomes the object for the second (right) lens

left lens:
- object distance: $u$
- image distance: $v^{'}$
right lens:
- object distance: $u^{'} = - v^{'}$
- image distance: $v$
let the outside medium be air ( $n_{1} = 1$ ) and lens be glass $n_{2} = n$

\begin{matrix} \frac{1}{u} + \frac{n}{v^{'}} = \frac{n - 1}{R_{1}} \\ \frac{n}{- v^{'}} + \frac{1}{v} = - \frac{n - 1}{R_{2}} \\ ⟹ \frac{1}{u} + \frac{1}{v} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}}) \end{matrix}

for light from infinity, $u \to \infty$ , the light converges at the focus of the lens, at focal length $v = f$

\frac{1}{f} = (n - 1) (\frac{1}{R_{1}} - \frac{1}{R_{2}})

this is called the lens maker's equation
if the focal length is known, the thin lens equation, which neglects the offset between the two surfaces:

\frac{1}{u} + \frac{1}{v} = \frac{1}{f}

$1 / f$ is called the 'power' of the lens, measured in units of 'dioptres' $[m^{- 1}]$